Hey,
1) the baby can be Turkish citizens even if there is no wedlocks. The only requirement is recognition from his father.
2) You can stay for a longer period, however you should also apply for permit of residence.
3)The baby needs recognition from his father to become a turkish citizen
4) The answer is yes and no at the same time. The process will be faster for this case; however, he would have to pay compansetion, because his wife are gonna get hurt and the judge would have considered this for alimony and damages for mental anguish.
Best Regards